Asymptotic behavior of flows by powers of the Gaussian curvature

Abstract

We consider a one-parameter family of strictly convex hypersurfaces in ${\mathbb{R}}^{n+1}$ moving with speed $-K^{\alpha }\nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha \ge \frac{1}{n+2}$. For $\alpha >\frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha =\frac{1}{n+2}$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

arXiv commentfinal version, to appear in Acta Math